#93624

order	512
nilpotency class	4
isoclinism	isoclinic class of #93517
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#340
commutator	C₂²×C₈	C₄²
Frattini, cosocle	C₂²×C₄×C₈	C₂²
prime radical	C₂⁴	#107
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

250 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 2 generators on 40 points:

(1,2)(3,5,7,8)(4,6)(9,10,12,17,14,18,13,16)(11,19,20,15,22,23,21,24)(25,26)(27,31)(28,35)(29,34)(30,33)(32,37)(36,40)(38,39);(1,3)(2,5,6,8)(4,7)(9,11,13,21,14,22,12,20)(10,15,17,23,18,24,16,19)(26,31,33,39)(27,36)(28,29)(32,35,40,34)(37,38)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fhi,d^2=fg,e^2=fg,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=efghi,f^a=fi,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 820);